Natural Monopoly and Price Regulation: P = MC vs P = ATC

Picture a river town with one toll bridge. A second bridge right beside it would make each crossing cost more to provide, not less. So why does competition backfire here? And with one bridge, what stops the owner from charging any toll it likes? This is Module 4.2, part B: natural monopoly, regulation and policy.

The bridge's loan payments and upkeep cost $12,000 a day, however many cars cross. That's a fixed cost. Each extra car adds only $2, for road wear and booth staff, so marginal cost, MC, is a flat $2.

Quick refresher from Module 3.2: average total cost, ATC, is average variable cost plus average fixed cost. Here, that's $2 plus $12,000 divided by the number of crossings.

At 3,000 crossings a day, ATC is $2 plus $4, or $6. At 6,000, it's $2 plus $2, or $4. More cars spread the fixed cost thinner. Say 6,000 cars cross each day. One bridge carries them all at $4 a crossing. Split them over two bridges, 3,000 each, and each owes the full $12,000: $6 a crossing.

So what just happened? One: the bridge has a big fixed cost, and each car adds only $2. Two: more crossings spread that cost, so ATC falls. Three: a second bridge would push the cost per crossing from $4 to $6.

That's a natural monopoly, built on economies of scale.

Quick refresher from Module 3.3: economies of scale means long-run average total cost falls as output rises.

A natural monopoly has economies of scale across the entire effective demand: ATC keeps falling over every quantity buyers might want, so one firm serves the market more cheaply than two. On the graph: axes Price, Cost and Quantity; demand, with MR below it reaching the quantity axis; MC flat at $2; and ATC sloping down across the whole graph. Exam trap: the flat MC is not what makes it natural. The reason is ATC falling over the entire effective demand.

Left alone, the owner sets the toll like any monopolist.

Quick refresher from Module 4.2, part A: produce where MR equals MC, then go straight up to demand for the price. MR equals $2 at 4,000 crossings, where demand is $6.

The unregulated bridge runs 4,000 crossings a day at a $6 toll, above its $2 MC. ATC there is $5.

Quick refresher from Unit 3: profit is price minus ATC, times quantity. $6 minus $5 is $1 a crossing, times 4,000: $4,000 a day.

Is that efficient? Demand meets MC at a $2 toll and 8,000 crossings.

Quick refresher from Module 3.7: output is allocatively efficient where price equals marginal cost, since demand shows what buyers value each unit at. So 8,000 is the socially optimal quantity.

Between 4,000 and 8,000 crossings, drivers value a trip above its $2 cost, but those trips never happen: that lost surplus is the deadweight loss.

Quick refresher from Unit 2: deadweight loss is a triangle, one half times base times height. Base: 8,000 minus 4,000. Height: $6 minus $2. One half times 4,000 times $4 is $8,000 a day.

So what just happened? One: MR equals MC at 4,000 crossings, and demand sets a $6 toll. Two: profit is $4,000 a day. Three: the efficient 8,000 crossings never happen, so the deadweight loss is $8,000 a day.

What can government do? Against an ordinary monopoly, antitrust policy can block it or break it up, making the market more competitive. But splitting a natural monopoly pushes ATC up. So governments either run a natural monopoly themselves or regulate its price. Price regulation means a binding price ceiling on the toll. The owner needn't cut the toll to get more cars, so the cap becomes its marginal revenue. Option one: cap the toll at $2, where demand meets MC. Drivers want 8,000 crossings, the socially optimal quantity. For an ordinary monopoly, a binding price ceiling where demand meets MC is the fix.

But at 8,000 crossings, ATC is $3.50, above the $2 toll. ATC minus price, times quantity: $1.50 times 8,000 is a loss of $12,000 a day. Exam trap: at the socially optimal price, a natural monopoly earns negative economic profit, because ATC is above the price. On a recent AP exam, about one student in three earned this point. In the long run, a loss like that would close the bridge. That's why a natural monopoly needs a lump-sum subsidy, here $12,000 a day, to produce the allocatively efficient quantity. Option two: cap the toll where demand meets ATC: $4, at 6,000 crossings. ATC is $4 there too, so this price gives zero economic profit.

Crossings rise from 4,000 to 6,000, and the toll falls from $6 to $4: the lowest toll that avoids a loss. Exam trap: zero profit is not zero deadweight loss. At 6,000 crossings, the $4 toll is still above the $2 MC, and output is still short of 8,000. The leftover deadweight loss is one half times 2,000 times $2, or $2,000 a day. Smaller than $8,000, but not gone.

So what just happened? One: a $2 cap reaches 8,000 crossings, but the owner loses $12,000 a day, so it needs a lump-sum subsidy. Two: a $4 cap gives zero profit at 6,000. Three: deadweight loss shrinks from $8,000 to $2,000, but stays.

Last question: what moves a monopolist's output? Only a change in MR or MC, since output sits where they meet. Card one: a factory across the river closes, so demand falls. D and MR shift left, the new MR meets MC at a smaller quantity, and crossings fall. Card two: the town pays the owner $1 per crossing, a per-unit subsidy. That lowers marginal cost: MC drops from $2 to $1. Now MR meets MC at 4,500 crossings, and the toll falls to $5.50.

Exam trap: 'output rises' isn't enough. Say how: the subsidy lowers MC, so MC meets MR at a larger quantity. On a recent AP exam, fewer than one student in three explained it. Card three: instead, the town charges a flat $3,000 daily license fee, a lump-sum tax: the same bill whether one car crosses or 10,000. That's a fixed cost, so MC and MR don't move. Crossings stay at 4,000 and the toll at $6; only profit falls, from $4,000 to $1,000 a day.

So what just happened? One: a fall in demand shifts D and MR left, so output falls. Two: a per-unit subsidy lowers MC, so crossings rise to 4,500. Three: a lump-sum tax changes neither MC nor MR, so only profit moves.

Exam trap: don't raise the toll to pass on a lump-sum tax. It doesn't change marginal cost, so price and quantity stay put. The same goes for a lump-sum subsidy.

Exam moment: two questions the way the AP exam asks them, then one free-response part.

Question 1. A natural monopoly has a constant marginal cost of $4. Unregulated, it sells 400 units at $12. Regulators set its price at $10, where demand crosses ATC, and it sells 500 units. Its straight-line demand crosses marginal cost at 800 units. What deadweight loss remains? A) $0 B) $700 C) $900 D) $1,600 E) $1,800

The answer is C. The leftover triangle runs from 500 to 800 units, between the $10 price and the $4 marginal cost. One half times 300 times $6 is $900. Option A, $0, ignores that price is still above MC. B is how much the loss shrank, D is the loss before regulation, and E forgets the one half.

Question 2. A town's only movie theater and an island's only ferry line are profit-maximizing monopolies. The government gives the theater a per-unit subsidy and charges the ferry line a lump-sum tax. What happens to each firm's profit-maximizing quantity in the short run? A) Theater: increases. Ferry: decreases. B) Theater: increases. Ferry: no change. C) Theater: no change. Ferry: decreases. D) Theater: no change. Ferry: no change. E) Theater: decreases. Ferry: increases.

The answer is B. The per-unit subsidy lowers the theater's MC, so MC meets MR at a larger quantity. The lump-sum tax is a fixed cost, so the ferry's MC, MR and quantity stay put. Option A: it treats the lump-sum tax like a per-unit tax. C flips which policy moves MC, D says neither moves it, and E has the subsidy cutting output.

Free response. The graph on screen is for a valley's only cable car company. Is this firm a natural monopoly? Explain.

To earn the point: say yes, because its average total cost falls over the entire effective demand for its product. ATC is still falling where demand reaches the quantity axis. The costly mistake: pointing to the flat MC line, or saying ATC falls without tying it to the entire demand. On a recent AP exam, fewer than one student in three earned this point.

So what just earned the point? One: say yes. Two: point to average total cost, not marginal cost. Three: say it falls over the entire effective demand.

Recap: a natural monopoly's ATC falls over the entire effective demand. A zero-profit price shrinks deadweight loss but doesn't end it; the efficient price needs a lump-sum subsidy. Only MR or MC changes move output. Next: Module 4.3, Price Discrimination.